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The document discusses image restoration techniques to recover degraded images. It describes modeling image degradation using a degradation function and additive noise. Common noise sources and models are explained, including Gaussian, Rayleigh, Erlang, exponential, uniform, and impulse noise. Spatial filtering techniques for noise removal are covered, such as mean, order-statistic (median, max, min), and adaptive filters. Adaptive median filters are discussed that vary the filter window size until the median pixel value is not an impulse value. The goal of image restoration is to apply the inverse of the degradation process to recover the original undamaged image.

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Image Restoration
11/11/2022 2 Image Restoration ► Image restoration: recover an image that has been degraded by using a prior knowledge of the degradation phenomenon. ► Model the degradation and applying the inverse process in order to recover the original image.
11/11/2022 3 A Model of Image Degradation/Restoration Process ►Degradation ▪ Degradation function H ▪ Additive noise ) , ( y x 
11/11/2022 4 A Model of Image Degradation/Restoration Process If is a process, then the degraded image is given in the spatial domain by ( , ) ( , ) ( , ) ( , ) H linear, position-invariant g x y h x y f x y x y  = +
11/11/2022 5 A Model of Image Degradation/Restoration Process The model of the degraded image is given in the frequency domain by ( , ) ( , ) ( , ) ( , ) G u v H u v F u v N u v = +
11/11/2022 6 Noise Sources ► The principal sources of noise in digital images arise during image acquisition and/or transmission ✓ Image acquisition e.g., light levels, sensor temperature, etc. ✓ Transmission e.g., lightning or other atmospheric disturbance in wireless network
11/11/2022 7 Noise Models (1) ► White noise ▪ The Fourier spectrum of noise is constant ► With the exception of spatially periodic noise, we assume ▪ Noise is independent of spatial coordinates ▪ Noise is uncorrelated with respect to the image itself
11/11/2022 8 Noise Models (2) ➢ Gaussian noise Electronic circuit noise, sensor noise due to poor illumination and/or high temperature ➢ Rayleigh noise Range imaging
11/11/2022 9 Noise Models (3) ➢ Erlang (gamma) noise: Laser imaging ➢ Exponential noise: Laser imaging ➢ Uniform noise: Least descriptive; Basis for numerous random number generators ➢ Impulse noise: quick transients, such as faulty switching
11/11/2022 10 Gaussian Noise (1) 2 2 ( ) /2 The PDF of Gaussian random variable, z, is given by 1 ( ) 2 z z p z e   − − = where, represents intensity is the mean (average) value of z is the standard deviation z z 
11/11/2022 11 Gaussian Noise (2) 2 2 ( ) /2 The PDF of Gaussian random variable, z, is given by 1 ( ) 2 z z p z e   − − = ▪ 70% of its values will be in the range ▪ 95% of its values will be in the range   ) ( ), (     + −   ) 2 ( ), 2 (     + −
11/11/2022 12 Rayleigh Noise 2 ( ) / The PDF of Rayleigh noise is given by 2 ( ) for ( ) 0 for z a b z a e z a p z b z a − −  −   =     2 The mean and variance of this density are given by / 4 (4 ) 4 z a b b    = + − =
11/11/2022 13 Erlang (Gamma) Noise 1 The PDF of Erlang noise is given by for 0 ( ) ( 1)! 0 for b b az a z e z p z b z a − −    = −     2 2 The mean and variance of this density are given by / / z b a b a  = =
11/11/2022 14 Exponential Noise The PDF of exponential noise is given by for 0 ( ) 0 for az ae z p z z a −   =    2 2 The mean and variance of this density are given by 1/ 1/ z a a  = =
11/11/2022 15 Uniform Noise The PDF of uniform noise is given by 1 for a ( ) 0 otherwise z b p z b a     = −    2 2 The mean and variance of this density are given by ( ) / 2 ( ) /12 z a b b a  = + = −
11/11/2022 16 Impulse (Salt-and-Pepper) Noise The PDF of (bipolar) impulse noise is given by for ( ) for 0 otherwise a b P z a p z P z b =   = =    If either or is zero, the impulse noise is called a b P P unipolar if , gray-level will appear as a light dot, while level will appear like a dark dot. b a b a 
11/11/2022 17
11/11/2022 18 Examples of Noise: Original Image
11/11/2022 19 Examples of Noise: Noisy Images(1)
11/11/2022 20 Examples of Noise: Noisy Images(2)
11/11/2022 21 Periodic Noise ► Periodic noise in an image arises typically from electrical or electromechanical interference during image acquisition. ► It is a type of spatially dependent noise ► Periodic noise can be reduced significantly via frequency domain filtering
11/11/2022 22 An Example of Periodic Noise
11/11/2022 23 Estimation of Noise Parameters (1) The shape of the histogram identifies the closest PDF match
11/11/2022 24 Estimation of Noise Parameters (2) Consider a subimage denoted by , and let ( ), 0, 1, ..., -1, denote the probability estimates of the intensities of the pixels in . The mean and variance of the pixels in : s i S p z i L S S = 1 0 1 2 2 0 ( ) and ( ) ( ) L i s i i L i s i i z z p z z z p z  − = − = = = −  
11/11/2022 25 Restoration in the Presence of Noise Only ̶ Spatial Filtering Noise model without degradation ( , ) ( , ) ( , ) and ( , ) ( , ) ( , ) g x y f x y x y G u v F u v N u v  = + = +
11/11/2022 26 Spatial Filtering: Mean Filters (1) Let represent the set of coordinates in a rectangle subimage window of size , centered at ( , ). xy S m n x y  ( , ) Arithmetic mean filter 1 ( , ) ( , ) xy s t S f x y g s t mn  =  Smooths local variation in an image; Noise is reduced as a result of blurring
11/11/2022 27 Spatial Filtering: Mean Filters (2) 1 ( , ) Geometric mean filter ( , ) ( , ) xy mn s t S f x y g s t    =        Generally, a geometric mean filter achieves smoothing comparable to the arithmetic mean filter, but it tends to lose less image detail in the process
11/11/2022 28 Spatial Filtering: Mean Filters (3) ( , ) Harmonic mean filter ( , ) 1 ( , ) xy s t S mn f x y g s t  =  It works well for salt noise, but fails for pepper noise. It does well also with other types of noise like Gaussian noise.
11/11/2022 29 Spatial Filtering: Mean Filters (4) 1 ( , ) ( , ) Contraharmonic mean filter ( , ) ( , ) ( , ) xy xy Q s t S Q s t S g s t f x y g s t +   =   Q is the order of the filter. It is well suited for reducing the effects of salt-and- pepper noise. Q>0 for pepper noise and Q<0 for salt noise.
11/11/2022 30 Spatial Filtering: Example (1)
11/11/2022 31 Spatial Filtering: Example (2)
11/11/2022 32 Spatial Filtering: Example (3) For contraharmonic filter, MUST know weather noise is Salt or pepper.
11/11/2022 33 Spatial Filtering: Order-Statistic Filters (1)   ( , ) Max filter ( , ) max ( , ) xy s t S f x y g s t  =   ( , ) Median filter ( , ) ( , ) xy s t S f x y median g s t  =   ( , ) Min filter ( , ) min ( , ) xy s t S f x y g s t  = Effective in the presence of both unipolar and bipolar noise Reduces pepper noise; finds brightest point in an image Reduces salt noise; finds darkest point in an image
11/11/2022 34 Spatial Filtering: Order-Statistic Filters (2)     ( , ) ( , ) Midpoint filter 1 ( , ) max ( , ) min ( , ) 2 xy xy s t S s t S f x y g s t g s t     = +     Combines order statistics and averaging Works best for randomly distributed noise e.g. Gaussian or unfiorm
11/11/2022 35 Spatial Filtering: Order-Statistic Filters (3)   ( , ) Alpha-trimmed mean filter 1 ( , ) ( , ) xy r s t S f x y g s t mn d  = −  We delete the / 2 lowest and the / 2 highest intensity values of ( , ) in the neighborhood . Let ( , ) represent the remaining - pixels. xy r d d g s t S g s t mn d • Useful with multiple types of noise, e.g. combination of salt and pepper and Gaussian noise. • if d = mn -1 , reduces to median filter
11/11/2022 36 Repeated median filtering blurs
11/11/2022 37 Max filter removes some dark pixels Min filter removes some light pixels
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11/11/2022 39 Spatial Filtering: Adaptive Filters (1) Adaptive filters The behavior changes based on statistical characteristics of the image inside the filter region defined by the mхn rectangular window. The performance is superior to that of the filters discussed
11/11/2022 40 Adaptive Filters: Adaptive, Local Noise Reduction Filters (1) 2 : local region The response of the filter at the center point (x,y) of is based on four quantities: (a) ( , ), the value of the noisy image at ( , ); (b) , the variance of the noise corrupti xy xy S S g x y x y   2 ng ( , ) to form ( , ); (c) , the local mean of the pixels in ; (d) , the local variance of the pixels in . L xy L xy f x y g x y m S S 
11/11/2022 41 Adaptive Filters: Adaptive, Local Noise Reduction Filters (2) 2 2 The behavior of the filter: (a) if is zero, the filter should return simply the value of ( , ). (b) if the local variance is high relative to ,the filter should return a value cl g x y     ose to ( , ); (c) if the two variances are equal, the filter returns the arithmetic mean value of the pixels in . xy g x y S
11/11/2022 42 Adaptive Filters: Adaptive, Local Noise Reduction Filters (3)   2 2 An adaptive expression for obtaining ( , ) based on the assumptions: ( , ) ( , ) ( , ) L L f x y f x y g x y g x y m    = − −
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Shortcomings of Median Filtering ►Works only if spatial density of impulse noise is not large ( Pa and Pb smaller than 0.2) ► Adaptive median filter works for large Pa and Pb ▪ Preserves details while smoothing non-impulse noise ▪ Median filter cannot do this. 11/11/2022 44
11/11/2022 45 Adaptive Filters: Adaptive Median Filters (1) min max med max The notation: minimum intensity value in maximum intensity value in median intensity value in intensity value at coordinates ( , ) maximum all xy xy xy xy z S z S z S z x y S = = = = = owed size of xy S
Intuition behind adaptive median filter ►Keep increasing window size until z_med is not an impulse, i.e. z_min < z_med < z_max ►When this happens check z_xy ▪ If z_xy is not an impulse output z_xy ▪ If z_xy is an impulse output z_med (since z_med is guaranteed not to be an impulse) 11/11/2022 46
11/11/2022 47 Adaptive Filters: Adaptive Median Filters (2) med min med max max The adaptive median-filtering works in two stages: Stage A: A1 = ; A2 = if A1>0 and A2<0, go to stage B Else increase the window size if window size , re z z z z S − −  med min max med peat stage A; Else output Stage B: B1 = ; B2 = if B1>0 and B2<0, output ; Else output xy xy xy z z z z z z z − −
11/11/2022 48 Adaptive Filters: Adaptive Median Filters (2) med min med max max The adaptive median-filtering works in two stages: Stage A: A1 = ; A2 = if A1>0 and A2<0, go to stage B Else increase the window size if window size , re z z z z S − −  med min max med peat stage A; Else output Stage B: B1 = ; B2 = if B1>0 and B2<0, output ; Else output xy xy xy z z z z z z z − − The median filter output is an impulse or not The processed point is an impulse or not
11/11/2022 49 Example: Adaptive Median Filters
11/11/2022 50 Periodic Noise Reduction by Frequency Domain Filtering The basic idea Periodic noise appears as concentrated bursts of energy in the Fourier transform, at locations corresponding to the frequencies of the periodic interference Approach A selective filter is used to isolate the noise
11/11/2022 51 Perspective Plots of Bandreject Filters
11/11/2022 52 A Butterworth bandreject filter of order 4, with the appropriate radius and width to enclose completely the noise impulses
11/11/2022 53 Perspective Plots of Notch Filters
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11/11/2022 55 Several interference components are present, the methods discussed in the preceding sections are not always acceptable because they remove much image information The components tend to have broad skirts that carry information about the interference pattern and the skirts are not always easily detectable.
11/11/2022 56 Optimum Notch Filtering It minimizes local variances of the restored estimated Procedure for restoration tasks in multiple periodic interference Isolate the principal contributions of the interference pattern Subtract a variable, weighted portion of the pattern from the corrupted image ( , ) f x y
11/11/2022 57 Optimum Notch Filtering: Step 1 Extract the principal frequency components of the interference pattern Place a notch pass filter at the location of each spike. ( , ) ( , ) ( , ) NP N u v H u v G u v =   1 ( , ) ( , ) ( , ) NP x y H u v G u v  − = 
11/11/2022 58 Optimum Notch Filtering: Step 2 (1) Filtering procedure usually yields only an approximation of the true pattern. The effect of components not present in the estimate of ( , ) can be minimized instead by subtracting from ( , ) a weighte x y g x y  d portion of ( , ) to obtain an estimate of ( , ): ( , ) ( , ) ( , ) ( , ) x y f x y f x y g x y w x y x y   = − One approach is to select ( , ) so that the variance of the estimate ( , ) is minimized over a specified neighborhood of every point ( , ). w x y f x y x y
11/11/2022 59 Optimum Notch Filtering: Step 2 (2) 2 2 The local variance of ( , ): 1 ( , ) ( , ) ( , ) (2 1)(2 1) a b s a t b f x y x y f x s y t f x y a b  =− =−   = + + −     + +  
11/11/2022 60 Optimum Notch Filtering: Step (3)   2 2 2 The local variance of ( , ): 1 ( , ) ( , ) ( , ) (2 1)(2 1) ( , ) ( , ) ( , ) 1 (2 1)(2 1) ( , ) ( , ) ( , ) ( 1 (2 1)(2 1) a b s a t b a b s a t b f x y x y f x s y t f x y a b g x s y t w x s y t x s y s a b g x y w x y x y g a b    =− =− =− =−   = + + −     + +   + + − + + + +   =     + + − −       = + +       2 , ) ( , ) ( , ) ( , ) ( , ) ( , ) a b s a t b x s y t w x y x s y s g x y w x y x y   =− =−   + + − + +       − −         Assume that w(x,y) remains essentially constant over the neighborhood gives the approximation w(x+s, y+t) = w(x,y)
11/11/2022 61 Optimum Notch Filtering: Step (4)   2 2 The local variance of ( , ): ( , ) ( , ) ( , ) 1 ( , ) (2 1)(2 1) ( , ) ( , ) ( , ) a b s a t b f x y g x s y t w x y x s y s x y a b g x y w x y x y    =− =−   + + − + +   =     + + − −         2 2 2 2 ( , ) To minimize ( , ) , 0 ( , ) for ( , ), the result is ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) x y x y w x y w x y g x y x y g x y x y w x y x y x y        =  − = −
11/11/2022 62 Optimum Notch Filtering: Example
11/11/2022 63 Optimum Notch Filtering: Example
11/11/2022 64 Optimum Notch Filtering: Example
11/11/2022 65 Optimum Notch Filtering: Example
11/11/2022 66 Linear, Position-Invariant Degradations   ( , ) ( , ) ( , ) g x y H f x y x y  = +
11/11/2022 67 Linear, Position-Invariant Degradations     An operator having the input-output relationship ( , ) ( , ) is said to be position invariant if ( , ) ( , ) for any ( , ) and any and . g x y H f x y H f x y g x y f x y       = − − = − −       1 2 1 2 1 2 is linear ( , ) ( , ) ( , ) ( , ) and are any two input images. H H af x y bf x y aH f x y bH f x y f f + = +
11/11/2022 68 Linear, Position-Invariant Degradations ( , ) ( , ) ( , ) f x y f x y d d          − − = − −         Assume for a moment that ( , ) 0 if is a linear operator, ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) x y H g x y H f x y H f x y d d H f x y d d f H x y d d                         − −   − −   − − = =   = − −     = − − = − −       Impulse response Superposition (or Fredholm) integral of the first kind
11/11/2022 69 Linear, Position-Invariant Degradations       Assume for a moment that ( , ) 0 if is a linear operator and position invariant, ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) x y H H x y h x y g x y H f x y f H x y d d f h x y d d                      − −   − − = − − = − − = = − − = − −     Convolution integral in 2-D
11/11/2022 70 Linear, Position-Invariant Degradations In the presence of additive noise, if is a linear operator and position invariant, ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) H g x y f h x y d d x y h x y f x y x y G u v H u v F u v N u v           − − = − − + = + = +  
11/11/2022 71 Estimating the Degradation Function ► Three principal ways to estimate the degradation function 1. Observation 2. Experimentation 3. Mathematical Modeling
11/11/2022 72 Mathematical Modeling (1) ► Environmental conditions cause degradation A model about atmospheric turbulence 2 2 5/6 ( ) ( , ) : a constant that depends on the nature of the turbulence k u v H u v e k − + =
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11/11/2022 74 Mathematical Modeling (2) ► Derive a mathematical model from basic principles E.g., An image blurred by uniform linear motion between the image and the sensor during image acquisition
11/11/2022 75 Mathematical Modeling (3) 0 0 Suppose that an image ( , ) undergoes planar motion, ( ) and ( ) are the time-varying components of motion in the - and -directions, respectively. The optical imaging process is perfect. T is th f x y x t y t x y   0 0 0 e duration of the exposure. The blurred image ( , ) ( , ) ( ), ( ) T g x y g x y f x x t y y t dt = − − 
11/11/2022 76 Mathematical Modeling (4)       0 0 0 2 ( ) 2 ( ) 0 0 0 2 ( ) 0 0 0 ( , ) ( ), ( ) ( , ) ( , ) ( ), ( ) ( ), ( ) T j ux vy T j ux vy T j ux vy g x y f x x t y y t dt G u v g x y e dxdy f x x t y y t dt e dxdy f x x t y y t e dxdy dt      − + − −   − + − −   − + − − = − − =   = − −       = − −                  0 0 0 0 2 ( ) ( ) 0 2 ( ) ( ) 0 ( , ) ( , ) T j ux t vy t T j ux t vy t F u v e dt F u v e dt   − + − + = =  
11/11/2022 77 Mathematical Modeling (4)   0 0 0 2 ( ) ( ) 0 0 2 ( ) 0 ( , ) Suppose that the image undergoes uniform linear motion in the -direction only, at a rate given by ( ) / . ( , ) T j ux t vy t T j ux t H u v e dt x x t at T H u v e dt   − + − = = =   2 / 0 sin( ) T j uat T j ua e dt T ua e ua     − − = = 
11/11/2022 78 Mathematical Modeling (5)   0 0 0 0 2 ( ) ( ) 0 Suppose that the image undergoes uniform linear motion in the -direction and -direction, at a rate given by ( ) / and ( ) / ( , ) T j ux t vy t x y x t at T y t bt T H u v e dt e  − + = = = =    2 [ ] / 0 ( ) sin ( ) ( ) T j ua vb t T j ua vb dt T ua vb e ua vb     − + − + = + + 
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11/11/2022 81 Inverse Filtering An estimate of the transform of the original image ( , ) ( , ) ( , ) G u v F u v H u v = ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) F u v H u v N u v F u v H u v N u v F u v H u v + = = +
11/11/2022 82 Inverse Filtering ( , ) ( , ) ( , ) ( , ) N u v F u v F u v H u v = + 1. We can't exactly recover the undegraded image because ( , ) is not known. 2. If the degradation function has zero or very small values, then the ratio ( , ) / ( , ) could easily dominate th N u v N u v H u v e estimate ( , ). F u v
11/11/2022 83 Inverse Filtering ( )2 /2 ( EXAMPLE The image in Fig. 5.25(b) was inverse filtered using the exact inverse of the degradation function that generated that image. That is, the degradation function is ( , ) k u M v H u v e − − + = 5/6 2 /2) , 0.0025 N k   −     =
11/11/2022 84 Inverse Filtering ( )2 /2 ( EXAMPLE The image in Fig. 5.25(b) was inverse filtered using the exact inverse of the degradation function that generated that image. That is, the degradation function is ( , ) k u M v H u v e − − + = 5/6 2 /2) 0.0025, 480. N k M N   −     = = = One approach is to limit the filter frequencies to values near the origin.
11/11/2022 85 A Butterworth lowpass function of order 10 The poor performance of direct inverse filtering in general
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11/11/2022 87 Minimum Mean Square Error (Wiener) Filtering ➢ N. Wiener (1942) ➢ Objective Find an estimate of the uncorrupted image such that the mean square error between them is minimized   2 2 ( ) e E f f = −
11/11/2022 88 Minimum Mean Square Error (Wiener) Filtering 2 2 The minimum of the error function is given in the frequency domain by the expression *( , ) ( , ) ( , ) ( , ) ( , ) | ( , ) | ( , ) *( , ) | ( , ) | ( , f f H u v S u v F u v G u v S u v H u v S u v H u v H u v S u v     =   +     = + 2 2 ( , ) ) / ( , ) 1 | ( , ) | ( , ) ( , ) | ( , ) | ( , ) / ( , ) f f G u v S u v H u v G u v H u v H u v S u v S u v            =   +    
11/11/2022 89 Minimum Mean Square Error (Wiener) Filtering 2 2 2 2 1 | ( , ) | ( , ) ( , ) ( , ) | ( , ) | ( , ) / ( , ) ( , ):degradation function *( , ): complex conjugate of ( , ) | ( , ) | *( , ) ( , ) ( , ) | ( , ) | power spectrum of the noise ( , f f H u v F u v G u v H u v H u v S u v S u v H u v H u v H u v H u v H u v H u v S u v N u v S u v     =   +     = = = 2 ) | ( , ) | power spectrum of the undegraded image F u v = =
11/11/2022 90 Minimum Mean Square Error (Wiener) Filtering 2 2 1 | ( , ) | ( , ) ( , ) ( , ) | ( , ) | is a specified constant. Generally, the value of K is chosen interactively to yield the best visual results. H u v F u v G u v H u v H u v K K   =   +  
11/11/2022 91 Minimum Mean Square Error (Wiener) Filtering
11/11/2022 92 Left: degradated image Middle: inverse filtering Right: Wiener filtering
11/11/2022 93 Some Measures (1) 1 1 2 0 0 1 1 2 0 0 Singal-to-Noise Ratio (SNR) | ( , ) | | ( , ) | This ratio gives a measure of the level of information bearing singal power to the level of noise power. M N u v M N u v F u v SNR N u v − − = = − − = = =  
11/11/2022 94 Some Measures (2) 2 1 1 0 0 1 1 2 0 0 1 1 2 0 0 Mean Square Error (MSE) 1 MSE= ( , ) ( , ) Root-Mean-Sqaure-Error (RMSE) ( , ) RMSE | ( , ) ( , ) | M N x y M N u v M N u v f x y f x y MN f x y f x y f x y − − = = − − = = − − = =   −   = −   
11/11/2022 95
11/11/2022 96 Constrained Least Squares Filtering
Constrained Least Squares 11/11/2022 97
11/11/2022 98 Constrained Least Squares Filtering 2 2 *( , ) ( , ) ( , ) | ( , ) | | ( , ) | ( , ) is the Fourier transform of the function 0 1 0 ( , ) 1 4 1 0 1 0 is a parameter H u v F u v G u v H u v P u v P u v p x y     =   +   −     = − −     −  
11/11/2022 99 Examples
11/11/2022 100 Geometric Mean Filter 1 2 2 2 *( , ) | ( , ) | ( , ) ( , ) | ( , ) | | ( , ) | ( , ) / ( , ) f H u v H u v F u v G u v H u v H u v S u v S u v     −     =       +         1: inverse filter =0: parametric Wiener filter =1/2: geometric mean filter    =
11/11/2022 101 Image Reconstruction from Projection ► Reconstruct an image from a series of projections X-ray computed tomography (CT) “Computed tomography is a medical imaging method employing tomography where digital geometry processing is used to generate a three-dimensional image of the internals of an object from a large series of two-dimensional X-ray images taken around a single axis of rotation.” http://en.wikipedia.org/wiki/Computed_tomography
11/11/2022 102 Backprojection “ In computed tomography or other imaging techniques requiring reconstruction from multiple projections, an algorithm for calculating the contribution of each voxel of the structure to the measured ray data, to generate an image; the oldest and simplest method of image reconstruction. “ http://www.medilexicon.com/medicaldictionary.php?t=9165
11/11/2022 103 Image Reconstruction: Introduction Soft, uniform tissue Uniform with higher absorption Tumor Intensity is proportional to absorption
11/11/2022 104 Image Reconstruction: Introduction
11/11/2022 105
11/11/2022 106
11/11/2022 109 Projections and the Radon Transform
11/11/2022 110 Projections and the Radon Transform ( , ) ( , ) ( cos sin ) j k k k j g f x y x y dxdy         − − = + −  
11/11/2022 111 Projections and the Radon Transform   ( , ) ( , ) ( cos sin ) f g f x y x y dxdy         − −  = = + −   ► Radon transform gives the projection (line integral) of f(x,y) along an arbitrary line in the xy-plane   1 1 0 0 ( , ) ( , ) ( cos sin ) M N x y f g f x y x y       − − = =  = = + − 
11/11/2022 112 Example: Using the Radon transform to obtain the projection of a circular region 2 2 2 ( , ) 0 otherwise A x y r f x y  +  =   ► Assume that the circle is centered on the origin of the xy-plane. Because the object is circularly symmetric, its projections are the same for all angles, so we just check the projection for 0  =
11/11/2022 113 Example: Using the Radon transform to obtain the projection of a circular region 2 2 2 2 2 2 2 2 2 2 ( , ) ( , ) ( cos sin ) ( , ) ( ) ( , ) ( , ) 2 | | r 0 o r r r r g f x y x y dxdy f x y x dxdy f y dy f y dy Ady A r                   − −   − −  − − − − − − − = + − = − = = = −  =        therwise     
11/11/2022 114 2 2 2 | | r ( ) 0 otherwise A r g      −  =   
11/11/2022 115 Sinogram: The Result of Radon Transform ► Sinogram: the result of Radon transform is displayed as an image with and as rectilinear coordinates  
11/11/2022 116 Image Reconstruction ( , ) ( cos sin , ) f x y g x y     = + 0 ( , ) ( , ) f x y f x y d    =  0 ( , ) ( , ) f x y f x y    = =  A back-projected image formed is referred to as a laminogram
11/11/2022 117 Examples: Laminogram
11/11/2022 118 The Fourier-Slice Theorem 2 For a given value of , the 1-D Fourier transform of a projection with respect to is ( , ) ( , ) j G w g e d         − − =  2 2 2 ( cos sin ) ( , ) ( , ) ( cos sin ) ( , ) ( cos sin ) ( , ) j j j x y G f x y x y e d dxdy f x y x y e d dxdy f x y e dxdy                     − − − −    − − − −   − + − − = + −   = + −     =        
11/11/2022 119 The Fourier-Slice Theorem   2 ( cos sin ) 2 ( ) cos , sin cos , sin ( , ) ( , ) ( , ) ( , ) ( cos , sin ) j x y j ux vy u w v w u w v w G f x y e dxdy f x y e dxdy F u v F w w               − + − −   − + − − = = = = =   =     = =     2 ( , ) ( , ) j G w g e d       − − =  Fourier-slice theorem: The Fourier tansform of a projection is a slice of the 2-D Fourier transform of the region from which the projection was obtained
11/11/2022 120 Illustration of the Fourier-slice theorem
Reconstruction methods 1. Simple : - Nearest Neighbor (zero order interpolation) - First order interpolation. 2. Radon inversión formula 3. Iterative methods
Transform reconstruction Polar Sampling: E.g.: 9 point DFT in each direction , 8 projections The samples are equally spaced 2  1  The goal is to find the FT values on the cartesian grid given the values at the polar coordinate system (red points) We can then apply one of the following interpolation methods: Zeroth order → Nearest Neighbor First order → Weighed Sum of neighbor samples (Average)
Radon Inversion Formula Recall that IDFT 2D of Fc(Ω1, Ω2) is: 1 1 2 2 1 2 1 2 1 2 1 2 2 1 ( , ) ( , ) 4 j t j t c c f t t F e e d d  + +    =−  =− =       We can express this expression using polar coordinates: i.e.: (Ω1, Ω2)→ (ω, θ).   2  1 
Radon Inversion Formula To accomplish this we need the jacobian: ( , ) ( , ) . .: ( , ) ( ( , ), ( , )) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) si x g u v y y h u v i e f x y dxdy f g u v h u v J dudv where x x u v x y x y y x J y y u v u v u v u v = = =          = = = −           
Radon Inversion Formula Therefore fc(t1, t2) will be: 1 2 ( , ) cos ( , ) . .: cos ( , ) cos ( , ) si g y h sen i e sen x y J sen u v                 = =  = = −  = = =  1 2 ( cos ) 1 2 2 0 1 ( , ) ( cos , ) 4 j t t sen c c f t t F sen e d d             + + − =  
Formula for inversion of Radon Then we have: 1 2 ( cos ) 1 2 2 0 1 ( , ) ( cos , ) 4 j t t sen c c f t t F sen e d d             + + − =   ( ) P   1 2 ( cos ) 1 2 2 0 1 ( , ) ( ) 4 j t t sen c f t t P e d d           + + − =   I
Formula de inversion de Radon The second integral is an IDFT: 1 2 ( cos ) 1 2 2 0 1 ( , ) ( ) 4 j t t sen c f t t P e d d           + + − =   I   1 2 ( ) ( ) ( ) ( ) cos G P IDFT G g t y t t t sen           = = = + ( ) : j t I G e d where     + − =  IDFT
Formula de inversion de Radon This gives us: 1 2 cos ( ) t t t sen I g t    = + = 1 2 ( cos ) I g t t sen    = + 1 2 1 2 2 0 1 ( , ) ( cos ) 4 c f t t g t t sen d       = +  Substituting in fc(t1, t2) we have:
Formula de inversion de Radon Finally: 1 2 1 2 2 0 1 ( , ) ( cos ) 4 c f t t g t t sen d       = +    1 2 ( ) ( ) ( ) ( , ) cos G P g t p t IDFT y t t t sen           =  =  = +
Radon Inversion Formula Summary:   ( ) ( ) ( , ) P d g t p t IDFT d dt t        + − =  = −  1- Find 2- Evaluate 3- Substitute in fc(t1, t2)   ( ) ( ) ( , ) P d g t p t IDFT d dt t        + − =  = −  ( , ) p t  1 2 1 2 2 0 1 ( , ) ( cos ) 4 c f t t g t t sen d       = + 
What is FT of 1/t 11/11/2022 131
Convolution Backprojection 2 t 1 t t  0 t 0 ( ) g t 0 1 t 0 2 t ( ) g t 0 0 1 2 0 ( , ) ( ) c f t t g t = 0 t 0 2 t 0 1 t 1 2 2 0 1 ( , ) ( ) 4 c f t t g t d     = 
11/11/2022 133 Reconstruction Using Parallel-Beam Filtered Backprojections 2 ( ) ( , ) ( , ) j ux vy f x y F u v e dudv    + − − =   2 2 ( cos sin ) 0 0 2 2 ( cos sin ) 0 0 Let cos , sin ,then , ( , ) ( cos , sin ) ( , ) j w x y j w x y u w v w dudv wdwd f x y F w w e wdwd G w e wdwd                  +  + = = = = =     2 ( cos sin ) 0 ( , ) | | ( , ) j w x y f x y w G w e dwd        + − =   ( , 180 ) ( , ) G w G w   + = −
11/11/2022 134 Reconstruction Using Parallel-Beam Filtered Backprojections 2 ( cos sin ) 0 2 0 cos sin ( , ) | | ( , ) | | ( , ) j w x y j w x y f x y w G w e dwd w G w e dw d                + −  − = + =   =         It’s not integrable Approach: Window the ramp so it becomes zero outside of a defined frequency interval. That is, a window band-limits the ramp filter.
11/11/2022 135 Hamming / Hann Widow 2 ( 1)cos 0 ( 1) ( ) 1 0 otherwise w c c w M h w M   + −   −  = −    0.54, the function is called the Hamming window 0.5,the function is called the Han window c c = =
11/11/2022 136 The Plot of Hamming Widow
11/11/2022 137 Filtered Backprojection The complete, filtered backprojection (to obtain the reconstructed image f(x,y) ) is described as follows: 1. Compute the 1-D Fourier transform of each projection 2. Multiply each Fourier transform by the filter function |w| which has been multiplied by a suitable (e.g., Hamming) window 3. Obtain the inverse 1-D Fourier transform of each resulting filtered transform 4. Integrate (sum) all the 1-D inverse transforms from step 3
11/11/2022 138 Examples: Filtered Backprojection
11/11/2022 139 Examples: Filtered Backprojection

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restoration_recon_chap5_GW.pdf

  • 2. 11/11/2022 2 Image Restoration ► Image restoration: recover an image that has been degraded by using a prior knowledge of the degradation phenomenon. ► Model the degradation and applying the inverse process in order to recover the original image.
  • 3. 11/11/2022 3 A Model of Image Degradation/Restoration Process ►Degradation ▪ Degradation function H ▪ Additive noise ) , ( y x 
  • 4. 11/11/2022 4 A Model of Image Degradation/Restoration Process If is a process, then the degraded image is given in the spatial domain by ( , ) ( , ) ( , ) ( , ) H linear, position-invariant g x y h x y f x y x y  = +
  • 5. 11/11/2022 5 A Model of Image Degradation/Restoration Process The model of the degraded image is given in the frequency domain by ( , ) ( , ) ( , ) ( , ) G u v H u v F u v N u v = +
  • 6. 11/11/2022 6 Noise Sources ► The principal sources of noise in digital images arise during image acquisition and/or transmission ✓ Image acquisition e.g., light levels, sensor temperature, etc. ✓ Transmission e.g., lightning or other atmospheric disturbance in wireless network
  • 7. 11/11/2022 7 Noise Models (1) ► White noise ▪ The Fourier spectrum of noise is constant ► With the exception of spatially periodic noise, we assume ▪ Noise is independent of spatial coordinates ▪ Noise is uncorrelated with respect to the image itself
  • 8. 11/11/2022 8 Noise Models (2) ➢ Gaussian noise Electronic circuit noise, sensor noise due to poor illumination and/or high temperature ➢ Rayleigh noise Range imaging
  • 9. 11/11/2022 9 Noise Models (3) ➢ Erlang (gamma) noise: Laser imaging ➢ Exponential noise: Laser imaging ➢ Uniform noise: Least descriptive; Basis for numerous random number generators ➢ Impulse noise: quick transients, such as faulty switching
  • 10. 11/11/2022 10 Gaussian Noise (1) 2 2 ( ) /2 The PDF of Gaussian random variable, z, is given by 1 ( ) 2 z z p z e   − − = where, represents intensity is the mean (average) value of z is the standard deviation z z 
  • 11. 11/11/2022 11 Gaussian Noise (2) 2 2 ( ) /2 The PDF of Gaussian random variable, z, is given by 1 ( ) 2 z z p z e   − − = ▪ 70% of its values will be in the range ▪ 95% of its values will be in the range   ) ( ), (     + −   ) 2 ( ), 2 (     + −
  • 12. 11/11/2022 12 Rayleigh Noise 2 ( ) / The PDF of Rayleigh noise is given by 2 ( ) for ( ) 0 for z a b z a e z a p z b z a − −  −   =     2 The mean and variance of this density are given by / 4 (4 ) 4 z a b b    = + − =
  • 13. 11/11/2022 13 Erlang (Gamma) Noise 1 The PDF of Erlang noise is given by for 0 ( ) ( 1)! 0 for b b az a z e z p z b z a − −    = −     2 2 The mean and variance of this density are given by / / z b a b a  = =
  • 14. 11/11/2022 14 Exponential Noise The PDF of exponential noise is given by for 0 ( ) 0 for az ae z p z z a −   =    2 2 The mean and variance of this density are given by 1/ 1/ z a a  = =
  • 15. 11/11/2022 15 Uniform Noise The PDF of uniform noise is given by 1 for a ( ) 0 otherwise z b p z b a     = −    2 2 The mean and variance of this density are given by ( ) / 2 ( ) /12 z a b b a  = + = −
  • 16. 11/11/2022 16 Impulse (Salt-and-Pepper) Noise The PDF of (bipolar) impulse noise is given by for ( ) for 0 otherwise a b P z a p z P z b =   = =    If either or is zero, the impulse noise is called a b P P unipolar if , gray-level will appear as a light dot, while level will appear like a dark dot. b a b a 
  • 18. 11/11/2022 18 Examples of Noise: Original Image
  • 19. 11/11/2022 19 Examples of Noise: Noisy Images(1)
  • 20. 11/11/2022 20 Examples of Noise: Noisy Images(2)
  • 21. 11/11/2022 21 Periodic Noise ► Periodic noise in an image arises typically from electrical or electromechanical interference during image acquisition. ► It is a type of spatially dependent noise ► Periodic noise can be reduced significantly via frequency domain filtering
  • 22. 11/11/2022 22 An Example of Periodic Noise
  • 23. 11/11/2022 23 Estimation of Noise Parameters (1) The shape of the histogram identifies the closest PDF match
  • 24. 11/11/2022 24 Estimation of Noise Parameters (2) Consider a subimage denoted by , and let ( ), 0, 1, ..., -1, denote the probability estimates of the intensities of the pixels in . The mean and variance of the pixels in : s i S p z i L S S = 1 0 1 2 2 0 ( ) and ( ) ( ) L i s i i L i s i i z z p z z z p z  − = − = = = −  
  • 25. 11/11/2022 25 Restoration in the Presence of Noise Only ̶ Spatial Filtering Noise model without degradation ( , ) ( , ) ( , ) and ( , ) ( , ) ( , ) g x y f x y x y G u v F u v N u v  = + = +
  • 26. 11/11/2022 26 Spatial Filtering: Mean Filters (1) Let represent the set of coordinates in a rectangle subimage window of size , centered at ( , ). xy S m n x y  ( , ) Arithmetic mean filter 1 ( , ) ( , ) xy s t S f x y g s t mn  =  Smooths local variation in an image; Noise is reduced as a result of blurring
  • 27. 11/11/2022 27 Spatial Filtering: Mean Filters (2) 1 ( , ) Geometric mean filter ( , ) ( , ) xy mn s t S f x y g s t    =        Generally, a geometric mean filter achieves smoothing comparable to the arithmetic mean filter, but it tends to lose less image detail in the process
  • 28. 11/11/2022 28 Spatial Filtering: Mean Filters (3) ( , ) Harmonic mean filter ( , ) 1 ( , ) xy s t S mn f x y g s t  =  It works well for salt noise, but fails for pepper noise. It does well also with other types of noise like Gaussian noise.
  • 29. 11/11/2022 29 Spatial Filtering: Mean Filters (4) 1 ( , ) ( , ) Contraharmonic mean filter ( , ) ( , ) ( , ) xy xy Q s t S Q s t S g s t f x y g s t +   =   Q is the order of the filter. It is well suited for reducing the effects of salt-and- pepper noise. Q>0 for pepper noise and Q<0 for salt noise.
  • 32. 11/11/2022 32 Spatial Filtering: Example (3) For contraharmonic filter, MUST know weather noise is Salt or pepper.
  • 33. 11/11/2022 33 Spatial Filtering: Order-Statistic Filters (1)   ( , ) Max filter ( , ) max ( , ) xy s t S f x y g s t  =   ( , ) Median filter ( , ) ( , ) xy s t S f x y median g s t  =   ( , ) Min filter ( , ) min ( , ) xy s t S f x y g s t  = Effective in the presence of both unipolar and bipolar noise Reduces pepper noise; finds brightest point in an image Reduces salt noise; finds darkest point in an image
  • 34. 11/11/2022 34 Spatial Filtering: Order-Statistic Filters (2)     ( , ) ( , ) Midpoint filter 1 ( , ) max ( , ) min ( , ) 2 xy xy s t S s t S f x y g s t g s t     = +     Combines order statistics and averaging Works best for randomly distributed noise e.g. Gaussian or unfiorm
  • 35. 11/11/2022 35 Spatial Filtering: Order-Statistic Filters (3)   ( , ) Alpha-trimmed mean filter 1 ( , ) ( , ) xy r s t S f x y g s t mn d  = −  We delete the / 2 lowest and the / 2 highest intensity values of ( , ) in the neighborhood . Let ( , ) represent the remaining - pixels. xy r d d g s t S g s t mn d • Useful with multiple types of noise, e.g. combination of salt and pepper and Gaussian noise. • if d = mn -1 , reduces to median filter
  • 36. 11/11/2022 36 Repeated median filtering blurs
  • 37. 11/11/2022 37 Max filter removes some dark pixels Min filter removes some light pixels
  • 39. 11/11/2022 39 Spatial Filtering: Adaptive Filters (1) Adaptive filters The behavior changes based on statistical characteristics of the image inside the filter region defined by the mхn rectangular window. The performance is superior to that of the filters discussed
  • 40. 11/11/2022 40 Adaptive Filters: Adaptive, Local Noise Reduction Filters (1) 2 : local region The response of the filter at the center point (x,y) of is based on four quantities: (a) ( , ), the value of the noisy image at ( , ); (b) , the variance of the noise corrupti xy xy S S g x y x y   2 ng ( , ) to form ( , ); (c) , the local mean of the pixels in ; (d) , the local variance of the pixels in . L xy L xy f x y g x y m S S 
  • 41. 11/11/2022 41 Adaptive Filters: Adaptive, Local Noise Reduction Filters (2) 2 2 The behavior of the filter: (a) if is zero, the filter should return simply the value of ( , ). (b) if the local variance is high relative to ,the filter should return a value cl g x y     ose to ( , ); (c) if the two variances are equal, the filter returns the arithmetic mean value of the pixels in . xy g x y S
  • 42. 11/11/2022 42 Adaptive Filters: Adaptive, Local Noise Reduction Filters (3)   2 2 An adaptive expression for obtaining ( , ) based on the assumptions: ( , ) ( , ) ( , ) L L f x y f x y g x y g x y m    = − −
  • 44. Shortcomings of Median Filtering ►Works only if spatial density of impulse noise is not large ( Pa and Pb smaller than 0.2) ► Adaptive median filter works for large Pa and Pb ▪ Preserves details while smoothing non-impulse noise ▪ Median filter cannot do this. 11/11/2022 44
  • 45. 11/11/2022 45 Adaptive Filters: Adaptive Median Filters (1) min max med max The notation: minimum intensity value in maximum intensity value in median intensity value in intensity value at coordinates ( , ) maximum all xy xy xy xy z S z S z S z x y S = = = = = owed size of xy S
  • 46. Intuition behind adaptive median filter ►Keep increasing window size until z_med is not an impulse, i.e. z_min < z_med < z_max ►When this happens check z_xy ▪ If z_xy is not an impulse output z_xy ▪ If z_xy is an impulse output z_med (since z_med is guaranteed not to be an impulse) 11/11/2022 46
  • 47. 11/11/2022 47 Adaptive Filters: Adaptive Median Filters (2) med min med max max The adaptive median-filtering works in two stages: Stage A: A1 = ; A2 = if A1>0 and A2<0, go to stage B Else increase the window size if window size , re z z z z S − −  med min max med peat stage A; Else output Stage B: B1 = ; B2 = if B1>0 and B2<0, output ; Else output xy xy xy z z z z z z z − −
  • 48. 11/11/2022 48 Adaptive Filters: Adaptive Median Filters (2) med min med max max The adaptive median-filtering works in two stages: Stage A: A1 = ; A2 = if A1>0 and A2<0, go to stage B Else increase the window size if window size , re z z z z S − −  med min max med peat stage A; Else output Stage B: B1 = ; B2 = if B1>0 and B2<0, output ; Else output xy xy xy z z z z z z z − − The median filter output is an impulse or not The processed point is an impulse or not
  • 50. 11/11/2022 50 Periodic Noise Reduction by Frequency Domain Filtering The basic idea Periodic noise appears as concentrated bursts of energy in the Fourier transform, at locations corresponding to the frequencies of the periodic interference Approach A selective filter is used to isolate the noise
  • 51. 11/11/2022 51 Perspective Plots of Bandreject Filters
  • 52. 11/11/2022 52 A Butterworth bandreject filter of order 4, with the appropriate radius and width to enclose completely the noise impulses
  • 55. 11/11/2022 55 Several interference components are present, the methods discussed in the preceding sections are not always acceptable because they remove much image information The components tend to have broad skirts that carry information about the interference pattern and the skirts are not always easily detectable.
  • 56. 11/11/2022 56 Optimum Notch Filtering It minimizes local variances of the restored estimated Procedure for restoration tasks in multiple periodic interference Isolate the principal contributions of the interference pattern Subtract a variable, weighted portion of the pattern from the corrupted image ( , ) f x y
  • 57. 11/11/2022 57 Optimum Notch Filtering: Step 1 Extract the principal frequency components of the interference pattern Place a notch pass filter at the location of each spike. ( , ) ( , ) ( , ) NP N u v H u v G u v =   1 ( , ) ( , ) ( , ) NP x y H u v G u v  − = 
  • 58. 11/11/2022 58 Optimum Notch Filtering: Step 2 (1) Filtering procedure usually yields only an approximation of the true pattern. The effect of components not present in the estimate of ( , ) can be minimized instead by subtracting from ( , ) a weighte x y g x y  d portion of ( , ) to obtain an estimate of ( , ): ( , ) ( , ) ( , ) ( , ) x y f x y f x y g x y w x y x y   = − One approach is to select ( , ) so that the variance of the estimate ( , ) is minimized over a specified neighborhood of every point ( , ). w x y f x y x y
  • 59. 11/11/2022 59 Optimum Notch Filtering: Step 2 (2) 2 2 The local variance of ( , ): 1 ( , ) ( , ) ( , ) (2 1)(2 1) a b s a t b f x y x y f x s y t f x y a b  =− =−   = + + −     + +  
  • 60. 11/11/2022 60 Optimum Notch Filtering: Step (3)   2 2 2 The local variance of ( , ): 1 ( , ) ( , ) ( , ) (2 1)(2 1) ( , ) ( , ) ( , ) 1 (2 1)(2 1) ( , ) ( , ) ( , ) ( 1 (2 1)(2 1) a b s a t b a b s a t b f x y x y f x s y t f x y a b g x s y t w x s y t x s y s a b g x y w x y x y g a b    =− =− =− =−   = + + −     + +   + + − + + + +   =     + + − −       = + +       2 , ) ( , ) ( , ) ( , ) ( , ) ( , ) a b s a t b x s y t w x y x s y s g x y w x y x y   =− =−   + + − + +       − −         Assume that w(x,y) remains essentially constant over the neighborhood gives the approximation w(x+s, y+t) = w(x,y)
  • 61. 11/11/2022 61 Optimum Notch Filtering: Step (4)   2 2 The local variance of ( , ): ( , ) ( , ) ( , ) 1 ( , ) (2 1)(2 1) ( , ) ( , ) ( , ) a b s a t b f x y g x s y t w x y x s y s x y a b g x y w x y x y    =− =−   + + − + +   =     + + − −         2 2 2 2 ( , ) To minimize ( , ) , 0 ( , ) for ( , ), the result is ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) x y x y w x y w x y g x y x y g x y x y w x y x y x y        =  − = −
  • 62. 11/11/2022 62 Optimum Notch Filtering: Example
  • 63. 11/11/2022 63 Optimum Notch Filtering: Example
  • 64. 11/11/2022 64 Optimum Notch Filtering: Example
  • 65. 11/11/2022 65 Optimum Notch Filtering: Example
  • 66. 11/11/2022 66 Linear, Position-Invariant Degradations   ( , ) ( , ) ( , ) g x y H f x y x y  = +
  • 67. 11/11/2022 67 Linear, Position-Invariant Degradations     An operator having the input-output relationship ( , ) ( , ) is said to be position invariant if ( , ) ( , ) for any ( , ) and any and . g x y H f x y H f x y g x y f x y       = − − = − −       1 2 1 2 1 2 is linear ( , ) ( , ) ( , ) ( , ) and are any two input images. H H af x y bf x y aH f x y bH f x y f f + = +
  • 68. 11/11/2022 68 Linear, Position-Invariant Degradations ( , ) ( , ) ( , ) f x y f x y d d          − − = − −         Assume for a moment that ( , ) 0 if is a linear operator, ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) x y H g x y H f x y H f x y d d H f x y d d f H x y d d                         − −   − −   − − = =   = − −     = − − = − −       Impulse response Superposition (or Fredholm) integral of the first kind
  • 69. 11/11/2022 69 Linear, Position-Invariant Degradations       Assume for a moment that ( , ) 0 if is a linear operator and position invariant, ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) x y H H x y h x y g x y H f x y f H x y d d f h x y d d                      − −   − − = − − = − − = = − − = − −     Convolution integral in 2-D
  • 70. 11/11/2022 70 Linear, Position-Invariant Degradations In the presence of additive noise, if is a linear operator and position invariant, ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) H g x y f h x y d d x y h x y f x y x y G u v H u v F u v N u v           − − = − − + = + = +  
  • 71. 11/11/2022 71 Estimating the Degradation Function ► Three principal ways to estimate the degradation function 1. Observation 2. Experimentation 3. Mathematical Modeling
  • 72. 11/11/2022 72 Mathematical Modeling (1) ► Environmental conditions cause degradation A model about atmospheric turbulence 2 2 5/6 ( ) ( , ) : a constant that depends on the nature of the turbulence k u v H u v e k − + =
  • 74. 11/11/2022 74 Mathematical Modeling (2) ► Derive a mathematical model from basic principles E.g., An image blurred by uniform linear motion between the image and the sensor during image acquisition
  • 75. 11/11/2022 75 Mathematical Modeling (3) 0 0 Suppose that an image ( , ) undergoes planar motion, ( ) and ( ) are the time-varying components of motion in the - and -directions, respectively. The optical imaging process is perfect. T is th f x y x t y t x y   0 0 0 e duration of the exposure. The blurred image ( , ) ( , ) ( ), ( ) T g x y g x y f x x t y y t dt = − − 
  • 76. 11/11/2022 76 Mathematical Modeling (4)       0 0 0 2 ( ) 2 ( ) 0 0 0 2 ( ) 0 0 0 ( , ) ( ), ( ) ( , ) ( , ) ( ), ( ) ( ), ( ) T j ux vy T j ux vy T j ux vy g x y f x x t y y t dt G u v g x y e dxdy f x x t y y t dt e dxdy f x x t y y t e dxdy dt      − + − −   − + − −   − + − − = − − =   = − −       = − −                  0 0 0 0 2 ( ) ( ) 0 2 ( ) ( ) 0 ( , ) ( , ) T j ux t vy t T j ux t vy t F u v e dt F u v e dt   − + − + = =  
  • 77. 11/11/2022 77 Mathematical Modeling (4)   0 0 0 2 ( ) ( ) 0 0 2 ( ) 0 ( , ) Suppose that the image undergoes uniform linear motion in the -direction only, at a rate given by ( ) / . ( , ) T j ux t vy t T j ux t H u v e dt x x t at T H u v e dt   − + − = = =   2 / 0 sin( ) T j uat T j ua e dt T ua e ua     − − = = 
  • 78. 11/11/2022 78 Mathematical Modeling (5)   0 0 0 0 2 ( ) ( ) 0 Suppose that the image undergoes uniform linear motion in the -direction and -direction, at a rate given by ( ) / and ( ) / ( , ) T j ux t vy t x y x t at T y t bt T H u v e dt e  − + = = = =    2 [ ] / 0 ( ) sin ( ) ( ) T j ua vb t T j ua vb dt T ua vb e ua vb     − + − + = + + 
  • 81. 11/11/2022 81 Inverse Filtering An estimate of the transform of the original image ( , ) ( , ) ( , ) G u v F u v H u v = ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) F u v H u v N u v F u v H u v N u v F u v H u v + = = +
  • 82. 11/11/2022 82 Inverse Filtering ( , ) ( , ) ( , ) ( , ) N u v F u v F u v H u v = + 1. We can't exactly recover the undegraded image because ( , ) is not known. 2. If the degradation function has zero or very small values, then the ratio ( , ) / ( , ) could easily dominate th N u v N u v H u v e estimate ( , ). F u v
  • 83. 11/11/2022 83 Inverse Filtering ( )2 /2 ( EXAMPLE The image in Fig. 5.25(b) was inverse filtered using the exact inverse of the degradation function that generated that image. That is, the degradation function is ( , ) k u M v H u v e − − + = 5/6 2 /2) , 0.0025 N k   −     =
  • 84. 11/11/2022 84 Inverse Filtering ( )2 /2 ( EXAMPLE The image in Fig. 5.25(b) was inverse filtered using the exact inverse of the degradation function that generated that image. That is, the degradation function is ( , ) k u M v H u v e − − + = 5/6 2 /2) 0.0025, 480. N k M N   −     = = = One approach is to limit the filter frequencies to values near the origin.
  • 85. 11/11/2022 85 A Butterworth lowpass function of order 10 The poor performance of direct inverse filtering in general
  • 87. 11/11/2022 87 Minimum Mean Square Error (Wiener) Filtering ➢ N. Wiener (1942) ➢ Objective Find an estimate of the uncorrupted image such that the mean square error between them is minimized   2 2 ( ) e E f f = −
  • 88. 11/11/2022 88 Minimum Mean Square Error (Wiener) Filtering 2 2 The minimum of the error function is given in the frequency domain by the expression *( , ) ( , ) ( , ) ( , ) ( , ) | ( , ) | ( , ) *( , ) | ( , ) | ( , f f H u v S u v F u v G u v S u v H u v S u v H u v H u v S u v     =   +     = + 2 2 ( , ) ) / ( , ) 1 | ( , ) | ( , ) ( , ) | ( , ) | ( , ) / ( , ) f f G u v S u v H u v G u v H u v H u v S u v S u v            =   +    
  • 89. 11/11/2022 89 Minimum Mean Square Error (Wiener) Filtering 2 2 2 2 1 | ( , ) | ( , ) ( , ) ( , ) | ( , ) | ( , ) / ( , ) ( , ):degradation function *( , ): complex conjugate of ( , ) | ( , ) | *( , ) ( , ) ( , ) | ( , ) | power spectrum of the noise ( , f f H u v F u v G u v H u v H u v S u v S u v H u v H u v H u v H u v H u v H u v S u v N u v S u v     =   +     = = = 2 ) | ( , ) | power spectrum of the undegraded image F u v = =
  • 90. 11/11/2022 90 Minimum Mean Square Error (Wiener) Filtering 2 2 1 | ( , ) | ( , ) ( , ) ( , ) | ( , ) | is a specified constant. Generally, the value of K is chosen interactively to yield the best visual results. H u v F u v G u v H u v H u v K K   =   +  
  • 91. 11/11/2022 91 Minimum Mean Square Error (Wiener) Filtering
  • 93. 11/11/2022 93 Some Measures (1) 1 1 2 0 0 1 1 2 0 0 Singal-to-Noise Ratio (SNR) | ( , ) | | ( , ) | This ratio gives a measure of the level of information bearing singal power to the level of noise power. M N u v M N u v F u v SNR N u v − − = = − − = = =  
  • 94. 11/11/2022 94 Some Measures (2) 2 1 1 0 0 1 1 2 0 0 1 1 2 0 0 Mean Square Error (MSE) 1 MSE= ( , ) ( , ) Root-Mean-Sqaure-Error (RMSE) ( , ) RMSE | ( , ) ( , ) | M N x y M N u v M N u v f x y f x y MN f x y f x y f x y − − = = − − = = − − = =   −   = −   
  • 96. 11/11/2022 96 Constrained Least Squares Filtering
  • 98. 11/11/2022 98 Constrained Least Squares Filtering 2 2 *( , ) ( , ) ( , ) | ( , ) | | ( , ) | ( , ) is the Fourier transform of the function 0 1 0 ( , ) 1 4 1 0 1 0 is a parameter H u v F u v G u v H u v P u v P u v p x y     =   +   −     = − −     −  
  • 100. 11/11/2022 100 Geometric Mean Filter 1 2 2 2 *( , ) | ( , ) | ( , ) ( , ) | ( , ) | | ( , ) | ( , ) / ( , ) f H u v H u v F u v G u v H u v H u v S u v S u v     −     =       +         1: inverse filter =0: parametric Wiener filter =1/2: geometric mean filter    =
  • 101. 11/11/2022 101 Image Reconstruction from Projection ► Reconstruct an image from a series of projections X-ray computed tomography (CT) “Computed tomography is a medical imaging method employing tomography where digital geometry processing is used to generate a three-dimensional image of the internals of an object from a large series of two-dimensional X-ray images taken around a single axis of rotation.” http://en.wikipedia.org/wiki/Computed_tomography
  • 102. 11/11/2022 102 Backprojection “ In computed tomography or other imaging techniques requiring reconstruction from multiple projections, an algorithm for calculating the contribution of each voxel of the structure to the measured ray data, to generate an image; the oldest and simplest method of image reconstruction. “ http://www.medilexicon.com/medicaldictionary.php?t=9165
  • 103. 11/11/2022 103 Image Reconstruction: Introduction Soft, uniform tissue Uniform with higher absorption Tumor Intensity is proportional to absorption
  • 107. 11/11/2022 109 Projections and the Radon Transform
  • 108. 11/11/2022 110 Projections and the Radon Transform ( , ) ( , ) ( cos sin ) j k k k j g f x y x y dxdy         − − = + −  
  • 109. 11/11/2022 111 Projections and the Radon Transform   ( , ) ( , ) ( cos sin ) f g f x y x y dxdy         − −  = = + −   ► Radon transform gives the projection (line integral) of f(x,y) along an arbitrary line in the xy-plane   1 1 0 0 ( , ) ( , ) ( cos sin ) M N x y f g f x y x y       − − = =  = = + − 
  • 110. 11/11/2022 112 Example: Using the Radon transform to obtain the projection of a circular region 2 2 2 ( , ) 0 otherwise A x y r f x y  +  =   ► Assume that the circle is centered on the origin of the xy-plane. Because the object is circularly symmetric, its projections are the same for all angles, so we just check the projection for 0  =
  • 111. 11/11/2022 113 Example: Using the Radon transform to obtain the projection of a circular region 2 2 2 2 2 2 2 2 2 2 ( , ) ( , ) ( cos sin ) ( , ) ( ) ( , ) ( , ) 2 | | r 0 o r r r r g f x y x y dxdy f x y x dxdy f y dy f y dy Ady A r                   − −   − −  − − − − − − − = + − = − = = = −  =        therwise     
  • 112. 11/11/2022 114 2 2 2 | | r ( ) 0 otherwise A r g      −  =   
  • 113. 11/11/2022 115 Sinogram: The Result of Radon Transform ► Sinogram: the result of Radon transform is displayed as an image with and as rectilinear coordinates  
  • 114. 11/11/2022 116 Image Reconstruction ( , ) ( cos sin , ) f x y g x y     = + 0 ( , ) ( , ) f x y f x y d    =  0 ( , ) ( , ) f x y f x y    = =  A back-projected image formed is referred to as a laminogram
  • 116. 11/11/2022 118 The Fourier-Slice Theorem 2 For a given value of , the 1-D Fourier transform of a projection with respect to is ( , ) ( , ) j G w g e d         − − =  2 2 2 ( cos sin ) ( , ) ( , ) ( cos sin ) ( , ) ( cos sin ) ( , ) j j j x y G f x y x y e d dxdy f x y x y e d dxdy f x y e dxdy                     − − − −    − − − −   − + − − = + −   = + −     =        
  • 117. 11/11/2022 119 The Fourier-Slice Theorem   2 ( cos sin ) 2 ( ) cos , sin cos , sin ( , ) ( , ) ( , ) ( , ) ( cos , sin ) j x y j ux vy u w v w u w v w G f x y e dxdy f x y e dxdy F u v F w w               − + − −   − + − − = = = = =   =     = =     2 ( , ) ( , ) j G w g e d       − − =  Fourier-slice theorem: The Fourier tansform of a projection is a slice of the 2-D Fourier transform of the region from which the projection was obtained
  • 118. 11/11/2022 120 Illustration of the Fourier-slice theorem
  • 119. Reconstruction methods 1. Simple : - Nearest Neighbor (zero order interpolation) - First order interpolation. 2. Radon inversión formula 3. Iterative methods
  • 120. Transform reconstruction Polar Sampling: E.g.: 9 point DFT in each direction , 8 projections The samples are equally spaced 2  1  The goal is to find the FT values on the cartesian grid given the values at the polar coordinate system (red points) We can then apply one of the following interpolation methods: Zeroth order → Nearest Neighbor First order → Weighed Sum of neighbor samples (Average)
  • 121. Radon Inversion Formula Recall that IDFT 2D of Fc(Ω1, Ω2) is: 1 1 2 2 1 2 1 2 1 2 1 2 2 1 ( , ) ( , ) 4 j t j t c c f t t F e e d d  + +    =−  =− =       We can express this expression using polar coordinates: i.e.: (Ω1, Ω2)→ (ω, θ).   2  1 
  • 122. Radon Inversion Formula To accomplish this we need the jacobian: ( , ) ( , ) . .: ( , ) ( ( , ), ( , )) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) si x g u v y y h u v i e f x y dxdy f g u v h u v J dudv where x x u v x y x y y x J y y u v u v u v u v = = =          = = = −           
  • 123. Radon Inversion Formula Therefore fc(t1, t2) will be: 1 2 ( , ) cos ( , ) . .: cos ( , ) cos ( , ) si g y h sen i e sen x y J sen u v                 = =  = = −  = = =  1 2 ( cos ) 1 2 2 0 1 ( , ) ( cos , ) 4 j t t sen c c f t t F sen e d d             + + − =  
  • 124. Formula for inversion of Radon Then we have: 1 2 ( cos ) 1 2 2 0 1 ( , ) ( cos , ) 4 j t t sen c c f t t F sen e d d             + + − =   ( ) P   1 2 ( cos ) 1 2 2 0 1 ( , ) ( ) 4 j t t sen c f t t P e d d           + + − =   I
  • 125. Formula de inversion de Radon The second integral is an IDFT: 1 2 ( cos ) 1 2 2 0 1 ( , ) ( ) 4 j t t sen c f t t P e d d           + + − =   I   1 2 ( ) ( ) ( ) ( ) cos G P IDFT G g t y t t t sen           = = = + ( ) : j t I G e d where     + − =  IDFT
  • 126. Formula de inversion de Radon This gives us: 1 2 cos ( ) t t t sen I g t    = + = 1 2 ( cos ) I g t t sen    = + 1 2 1 2 2 0 1 ( , ) ( cos ) 4 c f t t g t t sen d       = +  Substituting in fc(t1, t2) we have:
  • 127. Formula de inversion de Radon Finally: 1 2 1 2 2 0 1 ( , ) ( cos ) 4 c f t t g t t sen d       = +    1 2 ( ) ( ) ( ) ( , ) cos G P g t p t IDFT y t t t sen           =  =  = +
  • 128. Radon Inversion Formula Summary:   ( ) ( ) ( , ) P d g t p t IDFT d dt t        + − =  = −  1- Find 2- Evaluate 3- Substitute in fc(t1, t2)   ( ) ( ) ( , ) P d g t p t IDFT d dt t        + − =  = −  ( , ) p t  1 2 1 2 2 0 1 ( , ) ( cos ) 4 c f t t g t t sen d       = + 
  • 129. What is FT of 1/t 11/11/2022 131
  • 130. Convolution Backprojection 2 t 1 t t  0 t 0 ( ) g t 0 1 t 0 2 t ( ) g t 0 0 1 2 0 ( , ) ( ) c f t t g t = 0 t 0 2 t 0 1 t 1 2 2 0 1 ( , ) ( ) 4 c f t t g t d     = 
  • 131. 11/11/2022 133 Reconstruction Using Parallel-Beam Filtered Backprojections 2 ( ) ( , ) ( , ) j ux vy f x y F u v e dudv    + − − =   2 2 ( cos sin ) 0 0 2 2 ( cos sin ) 0 0 Let cos , sin ,then , ( , ) ( cos , sin ) ( , ) j w x y j w x y u w v w dudv wdwd f x y F w w e wdwd G w e wdwd                  +  + = = = = =     2 ( cos sin ) 0 ( , ) | | ( , ) j w x y f x y w G w e dwd        + − =   ( , 180 ) ( , ) G w G w   + = −
  • 132. 11/11/2022 134 Reconstruction Using Parallel-Beam Filtered Backprojections 2 ( cos sin ) 0 2 0 cos sin ( , ) | | ( , ) | | ( , ) j w x y j w x y f x y w G w e dwd w G w e dw d                + −  − = + =   =         It’s not integrable Approach: Window the ramp so it becomes zero outside of a defined frequency interval. That is, a window band-limits the ramp filter.
  • 133. 11/11/2022 135 Hamming / Hann Widow 2 ( 1)cos 0 ( 1) ( ) 1 0 otherwise w c c w M h w M   + −   −  = −    0.54, the function is called the Hamming window 0.5,the function is called the Han window c c = =
  • 134. 11/11/2022 136 The Plot of Hamming Widow
  • 135. 11/11/2022 137 Filtered Backprojection The complete, filtered backprojection (to obtain the reconstructed image f(x,y) ) is described as follows: 1. Compute the 1-D Fourier transform of each projection 2. Multiply each Fourier transform by the filter function |w| which has been multiplied by a suitable (e.g., Hamming) window 3. Obtain the inverse 1-D Fourier transform of each resulting filtered transform 4. Integrate (sum) all the 1-D inverse transforms from step 3